Functions and Graphs

Question 1

Relation

Answer 1

Any set of ordered pairs. The set of all first components of the ordered pairs is called the domain of the relation and the set of all second components is called the range of the relation.

Question 2

Function

Answer 2

A relation in which each member of the domain corresponds to exactly one member of the range.

Question 3

Vertical Line Test for Functions

Answer 3

If any vertical line intersects a graph in more than one point, the graph does not define y as a function of x.

Question 4

Zeros of a Function

Answer 4

The x-values for which f(x) = 0.

Question 5

Increasing Functions

Answer 5

Question 6

Decreasing Functions

Answer 6

Question 7

Constant Functions

Answer 7

Question 8

Relative Maximum

Answer 8

Question 9

Relative Minimum

Answer 9

Question 10

Symmetry and Tests for Symmetry

Answer 10

Question 11

Even Functions and Their Symmetries

Answer 11

The function f is an even function if

f( -x ) = f( x ) for all x in the domain of f.

The graph of an even function is symmetric with respect to the y-axis.

Question 12

Odd Functions and Their Symmetries

Answer 12

The function f is an even function if

f( -x ) = - f( x ) for all x in the domain of f.

The graph of an even function is symmetric with respect to the origin.

Question 13

Identifying Even or Odd Functions from Equations

Answer 13

• Even function: f( -x ) = f( x )
  The right side of the equation of an even function does not change if x is replaced with -x.
• Odd function: f( -x ) = -f( x )
  Every term on the right side of the equation of an odd function changes sign if x is replaced with -x.
• Neither even nor odd: f( -x ) <> f( x ) and f( -x ) <> -f( x )
  The right side of the equation changes if x is replaced with -x, but not every term on the right side changes sign.

Question 14

Piecewise Function

Answer 14

A function that is defined by two (or more) equations over a specified domain.

Question 15

Difference Quotient of a Function

Answer 15

Question 16

Slope

Answer 16

Question 17

Positive Slope

Answer 17

Question 18

Negative Slope

Answer 18

Question 19

Zero Slope

Answer 19

Question 20

Undefined Slope

Answer 20

Question 21

Point-Slope Form of the Equation of a Line

Answer 21

The point-slope form of the equation of a nonvertical line with slope m that passed through the point ( x₁ , y₁ ) is

y - y₁ = m ( x - x₁ )

Question 22

Slope-Intercept Form of the Equation of a Line

Answer 22

The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is

y = mx + b

Question 23

Graphing y = mx + b Using the Slope and y-intercept

Answer 23

1. Plot the point containing the y-intercept on the y-axis. This is the point ( 0 , b ).
2. Opbtain a second point using the slope, m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point.
3. Use a straightedge to draw a line through the two points. Draw arrowheads at the ends of teh line to show that the line continues indefinitely in both directions.

Question 24

Equation of a Horizontal Line

Answer 24

Question 25

Equation of a Vertical Line

Answer 25

Question 26

General Form of the Equation of a Line

Answer 26

Every Line has an equation that can be written in the general form Ax + By + C = 0, where A, B, and C are real numbers, and A and B are not both zero.

Question 27

Using Intercepts to Graph Ax + By + C = 0

Answer 27

1. Find the x-intercept. Let y = 0 and solve for x. Plot the point containing the x-intercept on the x-axis. 2. Find the y-intercept. Let x = 0 and solve for y. Plot the point containing the y-intercept on the y-axis. 3. Use a straightedge to draw a line through the two points containing the intercepts. Draw arrowheads at the ends of the line to show that the line continues indefinitely in both directions. As long as none of A, B, and C is zero, the graph of Ax + By + C = 0 will have distinct x- and y-intercepts, and this three step method an be used to graph the equation.

Question 28

Slope and Parallel Lines

Answer 28

1. If two nonvertical lines are parallel, then they have the same slope. 2. If two distinct nonvertical lines ahve the same slope, then they are parallel. 3. The two distinct vertical lines, both with undefined slopes, are parallel.

Question 29

Perpendicular

Answer 29

Two lines that intersect at a right angle (90 degrees)

Question 30

Slope and Perpendicular Lines

Answer 30

1. If two nonvertical lines are perpendicular, then the product of their slopes is -1. 2. If the product of the slopes of two lines is -1, then the lines are perpendicular. 3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.

Question 31

Average Rate of Change of a Function

Answer 31

Question 32

Vertical Shifts

Answer 32

Question 33

Horizontal Shifts

Answer 33

Question 34

Reflection about the x-Axis

Answer 34

The graph of y = -f( x ) is the graph of y = f( x ) reflected about the x-axis

Question 35

Reflection of the y-Axis

Answer 35

The graph of y = f( -x ) is the graph of y = f( x ) reflected about the y-axis

Question 36

Vertically Stretching and Shrinking Graphs

Answer 36

Question 37

Horizontally Stretching and Shrinking Graphs

Answer 37

Question 38

Summary of Transformations

Answer 38

Question 39

Order of Transformations

Answer 39

1. Horizontal Shifting 2. Stretching of Shrinking 3. Reflecting 4. Vertical Shifting

Question 40

Finding a Function's Domain

Answer 40

If a function f does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of f( x ) is a real number. Exclude from a function's domain real numbers that cause division by zero and real numbers that result in an even rood, such as a square root, of a negative number.

Question 41

The Algebra of Functions: Sum, Difference, Product, Quotient

Answer 41

Question 42

Composition of Functions

Answer 42

Question 43

Excluding Values from the Domain of (f _o g)( x ) = f( g ( x ) )

Answer 43

Question 44

Definition of the Inverse of a Function

Answer 44

Question 45

Finding the Inverse of a Function

Answer 45

Question 46

The Horizontal Line Test for Inverse Functions

Answer 46

Question 47

The Distance Formula

Answer 47

Question 48

The Midpoint Formula

Answer 48

Question 49

Definition of a Circle

Answer 49

Question 50

Standard Form of the Equation of a Circle

Answer 50

Question 51

The General Form of the Equation of a Circle

Answer 51